System and method for determining fault location

ABSTRACT

Apparatus and method for locating faults in wired drill pipe while drilling. In one embodiment, a fault location system includes a plurality of conductively coupled media sections, impedance measurement electronics, and a fault locator. Each media section includes conductive couplers on opposing ends of the media section, and conductive media connected to and communicatively coupling the conductive couplers. The impedance measurement electronics is configured to measure an input impedance of the media sections. The fault locator is configured to determine a propagation constant for the media sections, and to determine, as a function of the input impedance and the propagation constant, a location of a fault in the media sections.

CROSS-REFERENCE TO RELATED APPLICATION

The present application claims priority to U.S. Provisional Patent Application No. 61/693,937, filed on Aug. 28, 2012, entitled “System and Method for Determining Fault Location,” which is hereby incorporated herein by reference in its entirety.

BACKGROUND

While drilling a wellbore in subsurface formations it is advantageous for measurement and command information to be transferred between the surface and the drilling tools in a timely fashion. Some drilling systems employ a high-speed communication network including communication media embedded in the drill pipe to facilitate timely information transfer between surface and downhole systems. Such drill pipe, known as “wired drill pipe” (WDP) includes communicative couplers at each end of each pipe joint and the aforementioned communication media extending between the couplers.

A system employing WDP for communication may include hundreds of individual wired drill pipes connected in series. Repeater subs may be interspersed among the WDPs to extend communication range. If one WDP (or repeater sub) has an electrical fault, then the entire communication system may fail.

In one particularly problematic scenario, an intermittent fault occurs while drilling, but disappears as the drill string is removed from the borehole. Such intermittent faults may be due to downhole pressures, downhole temperatures, shocks, rotating and bending, or other environmental effects that are not present when the drill pipe is retracted from the wellbore. If the fault cannot be traced to within a few joints of WDP, then large sections of WDP may have to be replaced. For example, if the repeater subs are spaced apart by 500 meters, then an intermittent fault may only be locatable to within the 500 meter section below the lowest repeater sub known to be operational. This uncertainty in the location of the fault may require large numbers of WDP joints to be available on the drilling rig. Each failure might require 500 meters of drill pipe to be replaced. If the fault only occurs under drilling conditions, then it may be impossible to identify exactly which drill pipe is failing at the rig site. Therefore, it is desirable to locate an intermittent fault while drilling, that is—with the WDP in the borehole.

SUMMARY

A system and apparatus for locating faults in wired drill pipe while drilling are disclosed herein. In one embodiment, a method includes disposing a drill string comprising a plurality of wired drill pipes in a borehole. The input impedance of the wire drill pipes is measured while drilling. A propagation constant for the wire drill pipes is determined. Based on the input impedance, whether a fault in the wired drill pipe is an open circuit or a short circuit is determined.

In another embodiment, an apparatus for drilling a borehole in formations includes a drill string comprising a plurality of wired drill pipes, and a wired drill pipe fault monitor coupled to the wired drill pipes. The fault monitor includes an impedance measuring system configured to measure an input impedance of the wired drill pipes while drilling the borehole, and a fault locator. The fault locator is configured to determine a propagation constant for the wired drill pipes. The fault locator is also configured to determine, as a function of the input impedance and the propagation constant, a location of a fault in the wired drill pipes.

In one embodiment, a fault location system includes a plurality of conductively coupled media sections, impedance measurement electronics, and a fault locator. Each media section includes a length of conductive media and conductive couplers communicatively connected to opposing ends of the conductive media. The impedance measurement electronics is configured to measure an input impedance of the media sections. The fault locator is configured to determine a propagation constant for the media sections, and to analyze the input impedance and determine, as a function of the input impedance and the propagation constant, a location of a fault in the media sections.

In some embodiments, a channel characterization system includes a first calibration unit, a second calibration unit, a conductive medium coupling the first calibration unit to the second calibration unit, and a processor coupled to the first calibration unit and the second calibration unit. The first and second calibration units are configured to exchange characterization signals via the conductive medium, to measure amplitude and phase of the characterization signal received via the conductive medium from the other calibration unit, and to provide the amplitude and phase measurements to the processor. The processor is configured to determine a propagation constant of the conductive medium based on the measurements.

In yet other embodiments, a method for characterizing a communication channel includes splitting, by a first calibration unit, a calibration signal transmitted by a second calibration unit via a conductive medium connecting the first and second calibration units into a first two signals. A first of the first two signals is mixed by the first calibration unit with a first oscillator signal generated by the first calibration unit to produce a first mixed signal. A second of the first two signals is mixed by the first calibration unit with a second oscillator signal generated by the first calibration unit to produce a second mixed signal. The first and second oscillator signals generated by the first calibration unit have a same frequency and quadrature phase offset. A sum of the first of the first two signals and the first oscillator signal is filtered from the first mixed signal to produce a first filtered signal. A sum of the second of the first two signals and the second oscillator signal is filtered from the second mixed signal to produce a second filtered signal. The first filtered signal is integrated over time to generate a first integrated signal. The second filtered signal is integrated over time to generate a second integrated signal. A propagation constant for the conductive medium is computed based on the first and second integrated signals.

BRIEF DESCRIPTION OF THE DRAWINGS

For a detailed description of exemplary embodiments of the invention, reference is now be made to the figures of the accompanying drawings. The figures are not necessarily to scale, and certain features and certain views of the figures may be shown exaggerated in scale or in schematic form in the interest of clarity and conciseness.

FIG. 1 shows a drilling system that includes wired drill pipe and wired drill pipe fault location in accordance with principles disclosed herein;

FIG. 2 shows a longitudinal cross-section of a conductively coupled pair of wired drill pipes in accordance with principles disclosed herein;

FIG. 3 shows a block diagram of a wired drill pipe fault monitoring system in accordance with principles disclosed herein;

FIG. 4 shows a schematic diagram of a wired drill pipe impedance measurement system in accordance with principles disclosed herein;

FIG. 5 shows a transmission line model of wired drill pipe in accordance with principles disclosed herein;

FIGS. 6A and 6B show a block diagrams of a channel characterization system including a pair of repeater subs configured to determine the propagation constant of wired drill pipe connecting the repeater subs in accordance with various embodiments;

FIG. 7 shows a flow diagram for a method for determining the propagation constant for wired drill pipe in accordance with various embodiments;

FIG. 8 shows a flow diagram for a method for determining the location of a fault in wired drill pipe in accordance with principles disclosed herein;

FIG. 9A shows a schematic diagram of wired drill pipe cable and contacts for determining attenuation and phase velocity in accordance with principles disclosed herein;

FIG. 9B shows a graphical depiction of the characteristic impedance of a string of wired drill pipes;

FIG. 10 shows a flow diagram for a method for determining the distance to a fault in wired drill pipe in accordance with principles disclosed herein;

FIG. 11 shows a plot of real and imaginary parts of wired drill pipe impedance measurement for wired drill pipe including a short circuit in accordance with principles disclosed herein;

FIG. 12 shows a plot of real and imaginary parts of wired drill pipe impedance measurement for wired drill pipe including an open circuit in accordance with principles disclosed herein;

FIG. 13 shows normalized input impedance of wired drill pipe in the presence of a short located 100 meters from the fault monitor;

FIG. 14 shows inverted distances to the short versus frequency computed in accordance with principles disclosed herein;

FIG. 15 shows a flow diagram for another method for determining the distance to a fault in wired drill pipe in accordance with principles disclosed herein;

FIG. 16 shows normalized input impedance of wired drill pipe in the presence of a short located 500 meters from the fault monitor;

FIG. 17 shows identification of a zero-crossing in the imaginary part of the normalized impedance to determine distance to the fault in accordance with principles disclosed herein;

FIG. 18 shows a flow diagram for yet another method for determining the distance to a fault in wired drill pipe in accordance with principles disclosed herein;

FIG. 19 shows normalized input impedance of wired drill pipe in the presence of a short located 2000 meters from the fault monitor;

FIG. 20 shows normalized impedance data fit to impedance functions in accordance with principles disclosed herein.

NOTATION AND NOMENCLATURE

Certain terms are used throughout the following description and claims to refer to particular system components. As one skilled in the art will appreciate, companies may refer to a component by different names. This document does not intend to distinguish between components that differ in name but not function. In the following discussion and in the claims, the terms “including” and “comprising” are used in an open-ended fashion, and thus should be interpreted to mean “including, but not limited to . . . . ” Also, the term “couple” or “couples” is intended to mean either an indirect or direct connection. Thus, if a first device couples to a second device, that connection may be through direct engagement of the devices or through an indirect connection via other devices and connections. The recitation “based on” means “based at least in part on.” Therefore, if X is based on Y, X may be based on Y and any number of other factors.

DETAILED DESCRIPTION

The following discussion is directed to various illustrative embodiments of the invention. The embodiments disclosed are not to be interpreted, or otherwise used, to limit the scope of the disclosure, including the claims. In addition, one skilled in the art will understand that the following description has broad application, and the discussion of any embodiment is meant only to be exemplary of that embodiment, and not intended to intimate that the scope of the disclosure, including the claims, is limited to that embodiment.

FIG. 1 shows a drilling system 100 that includes wired drill pipe (WDP) 118 and wired drill pipe fault location in accordance with principles disclosed herein. In the drilling system 100, a drilling platform 102 supports a derrick 104 having a traveling block 106 for raising and lowering a drill string 108. A kelly 110 supports the drill string 108 as it is lowered through a rotary table 112. In some embodiments, a top drive is used to rotate the drill string 108 in place of the kelly 110 and the rotary table 112. A drill bit 114 is positioned at the downhole end of the tool string 126, and is driven by rotation of the drill string 108 or by a downhole motor (not shown) positioned in the tool string 126 uphole of the drill bit 114. As the bit 114 rotates, it removes material from the various formations 136 and creates the borehole 116. A pump 120 circulates drilling fluid through a feed pipe 122 and downhole through the interior of drill string 108, through orifices in drill bit 114, back to the surface via the annulus 140 around drill string 108, and into a retention pit 124. The drilling fluid transports cuttings from the borehole 116 into the pit 124 and aids in maintaining the integrity of the borehole 116.

The drill string 108 includes a plurality of lengths (or joints) of wired drill pipe 118 that are communicatively coupled end-to-end. A surface sub 130 communicatively couples the wired drill pipes 118 to surface processing systems, such as the drilling control/analysis computer 128. The drill string 108 may also include a bottom hole assembly (BHA) interface 134 and repeater subs 132. The BHA interface 134 communicatively couples the WDP 118 to the tool of the bottom hole assembly. The repeater subs 132 are interspersed among with the wired drill pipes 118, and may boost the WDP signal transmitted through the drill string 108. The spacing between the repeater subs 132 may be related to the efficiency (i.e. attenuation) of the wired drill pipes 118. The lower the attenuation, the greater the distance (e.g., the number of WDP joints) between repeater subs 132. Repeater subs 132 may be individually addressable, so that a command can be sent from the surface computer 128 to a selected repeater sub 132. In response to the command, the selected repeater sub 132 may transmit an acknowledgement to the surface computer 128. Such individual addressability and command/response protocol can be used to verify that the WDPs 118 (i.e., the WDP system) are working correctly between the surface and the selected repeater sub 132.

FIG. 2 shows a longitudinal cross-section of a mated pair of wired drill pipes 118 (or a sub 130, 132, 134 and a WDP 118) in accordance with principles disclosed herein Each WDP 118 includes a communicative medium 202 (e.g., a coaxial cable, twisted pair, etc.) structurally incorporated or embedded over the length of the pipe 118, and an interface 206 at each end of the pipe 118 for communicating with an adjacent pipe 118, sub, or other component. The communicative medium 202 is connected to each interface 206. In some embodiments, the interface 206 may include a conductive contact 204 (e.g., an annular conductive contact) for forming a conductive connection with the adjacent component. The conductive contact 204 may be embedded in insulating material (i.e., insulator) 208. For example, FIG. 2 shows a pin end 210 of a first wired drill pipe 118 mated to a box end 212 of a second wired drill pipe 118 such that conductive contacts 204 of the wired drill pipes 118 electrically connect the cables 202 of the two wired drill pipes 118. The high bandwidth of the wired drill pipes 118 allows for transfers of large quantities of data at a high transfer rate.

The cable 202 that links the two ends of the wired drill pipe 118 has well defined electrical properties, such as characteristic impedance, phase velocity, and attenuation that may be relatively independent of frequency. Because the contacts 204 are short compared to the cable 202, the reactive effects of the contacts 204 are small. The contacts 204 modify the properties of WDP 118 such that there are only relatively benign effects on the WDP transmission line properties.

The contacts 204 can affect the WDP transmission properties through hard failures or soft failures. Common hard failure modes for WDPs 118 include an open circuit and a short circuit. An open circuit may be due to a gap between the two contacts 204, a break in the cable 202, or a bad connection between the cable 202 and the contact 204. An open circuit is represented by a high equivalent load impedance (e.g., thousands of ohms). A short circuit may be due to mechanical failure of the insulation 208 between the contact 204 and the drill pipe 118, a mechanical failure of the connection between the contact 204 and the cable 202, or by a metal wire or metal flake bridging the insulation between the contact 204 and the shoulder of the drill pipe 118. A short circuit is represented by a low equivalent load impedance (e.g., zero ohms). Such hard failures may be induced by harsh downhole conditions. An intermittent open circuit caused by shock is a common type of fault in the WDPs 118.

Soft failures can cause excessive attenuation that results in lost communication. Soft failures may be caused by poor electrical connection between contacts 204 of adjacent WDPs 118, or due to a conductive path between a contact 204 and the drill pipe 118. High contact resistance between two contacts 204 might result from corrosion, dried mud, lost circulation material, sand, or other debris on the faces of the contacts 204. A low shunt resistance between the contact 204 and the drill pipe 118 might result from conductive drilling fluids and gaps between the insulators 208.

Embodiments of the drilling system 100 are configured to precisely locate faults in the wired drill pipes 118 of the drill string 108. FIG. 3 shows a block diagram of a wired drill pipe fault monitoring system 300 in accordance with principles disclosed herein. The fault monitor 300 may be disposed in whole or in part in the repeaters subs 132, the surface sub 130, and/or the BHA interface 134 for locating faults in the joints of wired drill pipe 118 uphole or downhole of the fault monitor 300. In some embodiments, the surface computer 128 may implement a portion of the fault monitor 130. Because at least some portion of the fault monitor 300 may be replicated in the repeater subs 132, the BHA interface 134 and the surface sub 130, embodiments of the drilling system 100 can locate a fault in wired drill pipes 118 from two directions, thereby improving fault location accuracy. Embodiments of the fault monitoring system 300 locate a fault to within a few drill pipes 118. Thus, embodiments require that only a few drill pipes be removed from the drill string, thereby reducing the time and expense associated with correcting a fault in wired drill pipe 118.

Embodiments are also applicable to locating faults in various other types of conductive communication systems that employ multiple sections of conductively coupled media with conductive couplers disposed at terminal ends of each section.

The fault monitor 300 includes WDP interface 302, impedance measurement system 304, and fault locator 306. The WDP interface 302 connects the impedance measurement system 304 to the cable 202 and/or the contacts 204 of the sub including the fault monitor 300 (e.g., the repeater sub 132). In some embodiments, the WDP interface 302 may selectively and/or periodically connect the impedance measurement system 304 to the cable 202 and/or the contacts 204 via, for example, switches or relays. In other embodiments, the WDP interface 302 may fixedly connect the impedance measurement system 304 to the cable 202 and/or the contacts,

The impedance measurement system 304 includes electronic circuitry that measures the impedance of a section of wired drill pipes 118 connected to, and either uphole or downhole of, the fault monitor 300. FIG. 4 shows a schematic diagram of a wired drill pipe impedance measurement system 304 in accordance with principles disclosed herein. The WDP impedance measurement system 304 includes a signal generator 402, a resistor 404, and one or more vector voltmeters 406.

The signal generator 402 produces an oscillating signal of frequency f, and angular frequency ω=2πf. The signal generator 402 may produce frequencies over the entire transmission bandwidth of the WDP 118. The section of WDPs 118 driven by the impedance measurement system 304 is assumed to have a characteristic impedance z(ω), and is terminated by a load impedance Z_(t)(ω). The impedance measurement system 304 determines the amplitude and phase of the current injected into the WDP 118, I_(IN)(ω) from the voltage V_(R) across the resistor 404 (R), via I_(IN)(ω)=V_(R)(ω)/R. The voltage input to the WDP section is V_(IN). Both V_(R) and V_(IN) may be measured using the vector voltmeters 406. The input impedance can be obtained from:

Z _(IN)(ω)=V _(IN)(ω)/I _(IN)(ω)   (1)

When the section of WDP 118 is terminated by a load with the same impedance as the WDP characteristic impedance, i.e., Z_(t)(ω)=Z(ω), the input impedance is given by Z_(IN)(ω)=Z(ω). Hence, when the WDP system is operating correctly, the WDP impedance Z(ω) is obtained from measuring Z_(IN)(ω) with the impedance measurement system 304.

The fault monitor 300 may measure Z_(IN)(ω) periodically during drilling for at least two reasons. First, if the input impedance is unchanged and equal to that expected for WDPs 118, then the WDP system is functioning correctly. Accordingly, the values of Z_(IN)(ω) should be recorded over the telemetry bandwidth for future reference. Second, if the input impedance begins to significantly change, then the properties of the WDP system are being adversely affected by downhole conditions. Such change in impedance is an indication of a developing problem. If the telemetry signal becomes noisy, is intermittent, or fails altogether, then there is a fault somewhere in the WDPs 118.

The fault locator 306 collects the impedance measurements provided by the impedance measurement system 304, determines, based on the measurements and other indications of telemetry problems (e.g., discontinuation of communication with other repeater subs, etc.), whether a fault is present in the section of WDPs 118 adjacent to the fault monitor 300, and determines a location of the fault. The fault locator 306 includes processor(s) 308 and storage 310. The processor(s) 308 may include, for example, one or more general-purpose microprocessors, digital signal processors, microcontrollers, or other suitable instruction execution devices known in the art. Processor architectures generally include execution units (e.g., fixed point, floating point, integer, etc.), storage (e.g., registers, memory, etc.), instruction decoding, peripherals (e.g., interrupt controllers, timers, direct memory access controllers, etc.), input/output systems (e.g., serial ports, parallel ports, etc.) and various other components and sub-systems.

The storage 310 is a non-transitory computer-readable storage device and includes volatile storage such as random access memory, non-volatile storage (e.g., a hard drive, an optical storage device (e.g., CD or DVD), FLASH storage, read-only-memory), or combinations thereof. The storage 310 includes impedance measurements 314, propagation constant logic 312, and fault distance evaluation logic 316 and various data processed by and produced by the processor(s) 308. The impedance measurements 314 include WDP impedance values generated by the impedance measurement system 304. The propagation constant logic 312 includes instructions for determining a propagation constant value useable for determining the location of a fault in the WDPs 118. The fault distance evaluation logic 316 includes instructions for determining a distance from the fault monitor 300 to a fault in the WDPs 118 based on the impedance measurements and the propagation constant. Processors execute software instructions. Software instructions alone are incapable of performing a function. Therefore, any reference herein to a function performed by software instructions, or to software instructions performing a function is simply a shorthand means for stating that the function is performed by a processor executing the instructions.

FIG. 5 shows a transmission line model of wired drill pipes 118 in accordance with principles disclosed herein. If a fault develops at a point 502 at distance L from a measurement point 504 (e.g., the location of fault monitor 300). The fault can be represented as a terminating impedance, Z_(t), on the section of WDP 118 transmission line. (Note that while explicit dependence on angular frequency (ω) is not always stated herein, it is understood that the impedances are functions of frequency). If the fault is an open circuit, then Z_(t)>>Z. If the fault is a short circuit, then Z_(t)=0. The reflection coefficient at the location of the fault 502 is:

$\begin{matrix} {\Gamma \equiv \frac{Z_{t} - Z}{Z_{t} + Z}} & (2) \end{matrix}$

Three special cases are of particular interest: Γ=0 if Z_(t)=Z, Γ=−1 if Z_(t)=0, and Γ=1 if Z_(t)>>Z. The input impedance at location 504 in FIG. 5 is given by:

$\begin{matrix} {{Z_{IN}(\omega)} = {{Z(\omega)}\; \frac{1 + {{\Gamma exp}\left( {{- 2}{\gamma (\omega)}L} \right)}}{1 + {\Gamma \; {\exp \left( {{- 2}{\gamma (\omega)}L} \right)}}}}} & (3) \end{matrix}$

where γ(ω)=α(ω)+jβ(ω) is the complex propagation constant for WDP 118. The real part of the propagation constant α(ω) is related to the attenuation by:

Allen=8.686α,   (4)

and the imaginary part β(ω) is related to the phase velocity V_(p)(ω) and angular frequency by:

$\begin{matrix} {V_{P} = \frac{\omega}{\beta}} & (5) \end{matrix}$

or by:

β=ω/V _(p)(ω).   (6)

In general β is nearly a linear function of frequency, while a is relatively independent of frequency. The propagation constant γ can be determined in a variety of ways. For example, γ may be calculated from the known physical properties of the cable 202, the contacts 204, and the insulators 208. Alternatively, γ may be directly measured by transmitting a signal of known phase and amplitude at one end of a WDP 118 and measuring the phase and amplitude at the other end of the WDP 118. Such measurement may be performed at the surface using a network analyzer as is well known. A system and method for accurately measuring the propagation constant γ while the WDP 118 is in the borehole 116 is described herein. The fault monitor 300 determines γ(ω) as a function of frequency.

The normalized input impedance measured by the fault monitor 300 at point 504 is defined as:

$\begin{matrix} {{\zeta (\omega)} = {{{\zeta^{\prime}(\omega)} + {{j\zeta}^{''}(\omega)}} = {\frac{Z_{IN}(\omega)}{Z(\omega)} = {\frac{1 + {{\Gamma exp}\left( {{- 2}{\gamma (\omega)}L} \right)}}{1 - {{\Gamma exp}\left( {{- 2}{\gamma (\omega)}L} \right)}}.}}}} & (7) \end{matrix}$

Some embodiments of the drilling system 100 determine the propagation constant γ of the WDP 118 via a process that includes transmitting sinusoidal signals between two calibration subs coupled to the wired drill pipes 118. In some embodiments, the calibration subs may be included in the WDP repeater subs 132. The calibration subs measure the phase and amplitude of the sinusoidal signals propagated in both directions, and, based on the measurements, provide amplitude and phase information from which attenuation, phase velocity, and group velocity of the WDP 118 may be computed. In some embodiments, the calibration subs 602 may be disposed at opposing ends of the drill string 108 (e.g., a sub at the surface and a sub at the BHA 134, rather than included in WDP repeater subs 132.

FIGS. 6A and 6B show block diagrams of a pair of repeaters subs 132 (132A, 132B) configured to determine the propagation constant of the WDP 118 disposed between the repeater subs 132. That is, the repeater subs 132A, 132B include calibration subs 602, the blocks of which are shown in FIGS. 6A and 6B. Because the present technique for determination of the propagation constant includes transmission of sinusoidal signals from each the repeaters subs 132A, 132B to the other, the repeater subs 132A, 132B may include similar circuitry. In FIG. 6A, repeater sub 132A transmits sinusoidal signal to repeater sub 132B via WDP(s) 118, consequently, only a portion of the circuitry of repeater sub 132A is shown. Repeater sub 132B processes the received sinusoidal signal and produces information that can be used to determine channel parameters.

Each repeater sub 132A, 132B includes an oscillator 612, mixers 604 (604A, 604B), low pass filters 606 (606A, 606B), analog-to-digital converters 608 (608A, 608B), and a processor 610. In some embodiments, a single filter 606, digitizer 608, or other component may be shared by the two signal paths. The processor 610 may be remote from a repeater sub 132A, 132B in some embodiments. For example, the processor 610 may be disposed at the surface, and WDP channel characterization information may be transmitted to processor 610 at the surface by the repeater subs 132A, 132B via WDP telemetry. In some embodiments, the processor 610 may be included in the processor(s) 308.

The oscillator 612 provides a stable frequency source that allows the repeater sub 132A, 132B to generate a sinusoidal signal at frequencies of interest over the WDP transmission channel. In some embodiments, the oscillator 612 may be a dual-mode quartz oscillator suitable for downhole operation. Such oscillators may be accurate to 0.1 parts-per-million (ppm) and have a resolution of 0.2 ppb, and be qualified to 185° Celsius. Some embodiments may apply software correction to achieve even higher oscillator accuracy (e.g., 10 ppb to 40 ppb).

Characterization of the WDP channel between the repeater subs 132A, 132B includes measuring the propagation constant γ(ω)=α(ω)+jβ(ω) at a number of frequencies of interest over the bandwidth of the WDP channel. The imaginary part of γ is related to the phase velocity V_(p) via equation (6). The group velocity is related to β(ω) by

$V_{G} = {\frac{\partial\omega}{\partial\beta}.}$

The group velocity can be determined by measuring β at adjacent angular frequencies (ω,ω+dω) and computing

$\begin{matrix} {V_{G} \approx {\frac{\omega}{{\beta \left( {\omega + {\omega}} \right)} - {\beta (\omega)}}.}} & (8) \end{matrix}$

In the arrangement of FIG. 6A, repeater sub 132A generates a signal V₁ sin(w₁t+θ₁), where V₁ is a known voltage. For example, a voltmeter in the repeater sub 132A can measure the voltage V₁. The angular frequency ω₁ of the oscillator 612 is also known to a given accuracy. In some embodiments, the repeater sub 132A receives, via WDP telemetry, voltage and frequency parameters to apply in generating the signal, from a parameter source at the surface for example. If sub 132A is uphole of sub 1328 and the distance between the subs 132A, 132B is x with sub 132A located at x=0 and sub 132B located at x=L, then the downward propagating wave at any location x along the WDP 118 at time t is described by V₁e^(−αα) sin(ω₁t−βx+θ₁). The repeater subs 132A, 1328 may be sufficiently well matched to the WDP transmission line impedance that there are only negligible reflections.

The repeater sub 132B is configured to receive the signal transmitted by the sub 132A. The frequency of the oscillator 612 of the sub 132B is set to an angular frequency ω₂. Preferably, ω₂=ω₁ but there may be a small angular frequency difference Δω=ω₁−ω₂ where Δω<<ω₁,ω₂. The signal received at the repeater sub 132B is V₁e^(−αL) sin(ω₁t−βL+θ₁). The repeater sub 132B splits the received signal into two equal signals

$\frac{1}{2}V_{1}^{{- \alpha}\; L}{\sin \left( {{\omega_{1}t} - {\beta \; L} + \theta_{1}} \right)}$

and provides one of the two signals to each of the mixers 604A, 604B. The oscillator 612 of the sub 132B provides mixer 604A with a signal V sin(ω₂t+θ₂), and provides mixer 604B with a signal V cos(ω₂t+θ₂). Mixer 604A mixes

$\frac{1}{2}V_{1}^{{- \alpha}\; L}{\sin \left( {{\omega_{1}t} - {\beta \; L} + \theta_{1}} \right)}$

and V sin(ω₂t+θ₂) producing:

$\begin{matrix} {{{\rho_{1}(t)} = {\frac{1}{2}V_{1}^{{- \alpha}\; L}{\sin \left( {{\omega_{1}t} - {\beta \; L} + \theta_{1}} \right)}{\sin \left( {{\omega_{2}t} + \theta_{2}} \right)}}},} & (9) \\ {{\rho_{1}(t)} = {\frac{1}{4}V_{1}^{{- \alpha}\; L}{\begin{Bmatrix} {{\cos \left\lbrack {{\left( {\omega_{1} - \omega_{2}} \right)t} - {\beta \; L} + \theta_{1} - \theta_{2}} \right\rbrack} -} \\ {\cos \left\lbrack {{\left( {\omega_{1} + \omega_{2}} \right)t} - {\beta \; L} + \theta_{1} + \theta_{2}} \right\rbrack} \end{Bmatrix}.}}} & (10) \end{matrix}$

For simplicity, set V=1 volt.

The output of mixer 604A is provided to the low pass filter 606A. The low pass filter 606A blocks the high frequency term ω₁+ω₂ and passes the low frequency term Δω=ω₁−ω₂, producing signal:

$\begin{matrix} {{\rho_{2}(t)} = {\frac{1}{4}V_{1}^{{- \alpha}\; L}{\cos \left( {{\Delta \; \omega \; t} - {\beta \; L} + \theta_{1} - \theta_{2}} \right)}}} & (11) \end{matrix}$

Mixer 604B mixes

$\frac{1}{4}V_{1}^{{- \alpha}\; L}{\sin \left( {{\omega_{1}t} - {\beta \; L} + \theta_{1}} \right)}\mspace{14mu} {and}\mspace{14mu} V\; {\cos \left( {{\omega_{2}t} + \theta_{2}} \right)}$

producing:

$\begin{matrix} {{{\sigma_{1}(t)} = {\frac{1}{2}V_{1}^{{- \alpha}\; L}{\sin \left( {{\omega_{1}t} - {\beta \; L} + \theta_{1}} \right)}{\cos \left( {{\omega_{2}t} + \theta_{2}} \right)}}},} & (12) \\ {{\sigma_{1}(t)} = {\frac{1}{4}V_{1}^{{- \alpha}\; L}{\begin{Bmatrix} {{\sin \left\lbrack {{\left( {\omega_{1} - \omega_{2}} \right)t} - {\beta \; L} + \theta_{1} - \theta_{2}} \right\rbrack} +} \\ {\sin \left\lbrack {{\left( {\omega_{1} + \omega_{2}} \right)t} - {\beta \; L} + \theta_{1} + \theta_{2}} \right\rbrack} \end{Bmatrix}.}}} & (13) \end{matrix}$

The output of mixer 604B is provided to the low pass filter 606B. The low pass filter 606B blocks the high frequency term ω₁+ω₂ and passes the low frequency term Δω=ω₁−ω₂, producing signal:

$\begin{matrix} {{\sigma_{2}(t)} = {\frac{1}{4}V_{1}^{{- \alpha}\; L}{\sin \left( {{\Delta \; \omega \; t} - {\beta \; L} + \theta_{1} - \theta_{2}} \right)}}} & (14) \end{matrix}$

Signals ρ₂(t) and σ₂(t) are digitized by the A/D converters 608A and 608B, and the digitized signals are provided to the processor 610 for further processing.

Having acquired WDP characterization data using signal propagating in one direction along the WDP 118 (e.g., uphole to downhole), characterization data is acquired using signal propagating in the opposite direction along the WDP 118 (e.g., downhole to uphole). Thus, consider FIG. 6B where repeater sub 132B is downhole from repeater sub 132A and the signals ρ₂(t) and σ²(t) described above have been acquired by propagating signal from repeater sub 132A downhole to 132B. The oscillators 612 continue to operate at the same angular frequencies, ω₁ and ω₂ and with the same phases, θ₁ and θ₂. The repeater sub 132B generates the signal V₂ sin(ω₂t+θ₂). The voltage V₂ can be either set to a specific value or measured in the repeater sub 132B, and the voltage value digitally transmitted to the repeater sub 132A. The upward propagating wave on the WDP transmission line at any location x and any time t is V₂e^(α(x−L)) sin(ω₂t+β(x−L)+θ₂).

The signal received at the repeater sub 132A is V₂e^(−αL) sin(ω₂t−βL+θ₂). The repeater sub 132A splits the received signal into two equal signals

$\frac{1}{2}V_{2}^{{- \alpha}\; L}{\sin \left( {{\omega_{2}t} - {\beta \; L} + \theta_{2}} \right)}$

and provides one of the two signals to each of the mixers 604A, 604B. The oscillator 612 of the sub 132B provides mixer 604A with a signal V sin(ω₁t+θ₁), and provides mixer 604B with a signal V cos(ω₁t+θ₁), where V=1 volt for simplicity. Mixer 604A mixes

$\frac{1}{2}V_{2}^{{- \alpha}\; L}{\sin \left( {{\omega_{2}t} - {\beta \; L} + \theta_{2}} \right)}$

and V sin(ω₁t+θ₁) producing:

$\begin{matrix} {{{\delta_{1}(t)} = {\frac{1}{2}V_{2}^{{- \alpha}\; L}{\sin \left( {{\omega_{2}t} - {\beta \; L} + \theta_{2}} \right)}{\sin \left( {{\omega_{1}t} + \theta_{1}} \right)}}},} & (15) \\ {{\delta_{1}(t)} = {\frac{1}{4}V_{2}^{{- \alpha}\; L}{\begin{Bmatrix} {{\cos \left\lbrack {{\left( {\omega_{2} - \omega_{1}} \right)t} - {\beta \; L} + \theta_{2} - \theta_{1}} \right\rbrack} -} \\ {\cos \left\lbrack {{\left( {\omega_{1} + \omega_{2}} \right)t} - {\beta \; L} + \theta_{1} + \theta_{2}} \right\rbrack} \end{Bmatrix}.}}} & (16) \end{matrix}$

Mixer 604B mixes

$\frac{1}{2}V_{2}^{{- \alpha}\; L}{\sin \left( {{\omega_{2}t} - {\beta \; L} + \theta_{2}} \right)}$

and V cos(ω₁t+θ₁) producing:

$\begin{matrix} {{{ɛ_{1}(t)} = {\frac{1}{2}V_{2}^{{- \alpha}\; L}{\sin \left( {{\omega_{2}t} - {\beta \; L} + \theta_{2}} \right)}{\cos \left( {{\omega_{1}t} + \theta_{1}} \right)}}},} & (17) \\ {{ɛ_{1}(t)} = {\frac{1}{4}V_{2}^{{- \alpha}\; L}{\begin{Bmatrix} {{\sin \left\lbrack {{\left( {\omega_{2} - \omega_{1}} \right)t} - {\beta \; L} + \theta_{2} - \theta_{1}} \right\rbrack} +} \\ {\sin \left\lbrack {{\left( {\omega_{1} + \omega_{2}} \right)t} - {\beta \; L} + \theta_{1} + \theta_{2}} \right\rbrack} \end{Bmatrix}.}}} & (18) \end{matrix}$

The outputs of the mixers 604A, 604B are provided to the low pass filters 606A, 606B. From the mixer output data, the low pass filters 606A, 606B respectively produce

$\begin{matrix} {{\delta_{2}(t)} = {\frac{1}{4}V_{2}^{{- \alpha}\; L}{\cos \left( {{\Delta \; \omega \; t} - {\beta \; L} + \theta_{2} - \theta_{1}} \right)}\mspace{14mu} {and}}} & (19) \\ {{ɛ_{2}(t)} = {\frac{1}{4}V_{2}^{{- \alpha}\; L}{{\sin \left( {{\Delta \; \omega \; t} - {\beta \; L} + \theta_{2} - \theta_{1}} \right)}.}}} & (20) \end{matrix}$

Signals δ₂(1) and ε₂(t) are digitized by the ND converters 608A and 608B, and the digitized signals are provided to the processor 610 for further processing.

The instantaneous values ρ₂(t), σ₂(t), δ₂(t) and ε₂(t) are integrated using integration circuitry ahead of the ND converters 608A and 608B or by the processor 610 using a measurement time series. If a first repeater sub 132A is transmitting sinusoidal signal to a second repeater sub 132B during time t ∈ [−T,0], and the second repeater sub 132B is transmitting sinusoidal signal to a first repeater sub 132A during time t ∈ [0,T], then integration of each of ρ₂(t), δ₂(t), δ₂(t) and ε₂(t) produces:

$\begin{matrix} {{\rho_{3} = {\frac{1}{T}{\int_{- T}^{0}{{\rho_{2}(t)}{t}}}}},} & (21) \\ {{\sigma_{3} = {\frac{1}{T}{\int_{- T}^{0}{{\sigma_{2}(t)}{t}}}}},} & (22) \\ {{\delta_{3} = {\frac{1}{T}{\int_{0}^{T}{{\delta_{2}(t)}{t}}}}},{and}} & (23) \\ {ɛ_{3} = {\frac{1}{T}{\int_{0}^{T}{{ɛ_{2}(t)}{{t}.}}}}} & (24) \end{matrix}$

Embodiments may let φ₁≡θ₁−θ₂−βL, and set the variable of integration to u≡Δωt+φ₁, resulting in:

$\begin{matrix} {\rho_{3} = {{\frac{1}{T}{\int_{- T}^{0}{{t}\left\{ {\frac{1}{4}V_{1}^{{- \alpha}\; L}{\cos \left( {{\Delta \; \omega \; t} + \varphi_{1}} \right)}} \right\}}}} = {\frac{V_{1}^{{- \alpha}\; L}}{4\Delta \; \omega \; T}{\int_{\varphi_{1} - {\Delta \; \omega \; T}}^{\varphi_{1}}{{u}\left\{ {\cos \; u} \right\}}}}}} & (25) \\ {\rho_{3} = {{\frac{V_{1}^{{- \alpha}\; L}}{4\Delta \; \omega \; T}\left\{ {{\sin \; \varphi_{1}} - {\sin \left( {\varphi_{1} - {\Delta \; \omega \; T}} \right)}} \right\}} = {\frac{V_{1}^{{- \alpha}\; L}}{4\Delta \; \omega \; T}\left\{ {2{\cos \left( {\varphi_{1} - {\Delta \; \omega \; {T/2}}} \right)}{\sin \left( {\Delta \; \omega \; {T/2}} \right)}} \right\}}}} & (26) \\ {\mspace{20mu} {\rho_{3} = {\frac{V_{1}^{{- \alpha}\; L}}{4}{{\cos \left( {\varphi_{1} - {\Delta \; \omega \; {T/2}}} \right)}\left\lbrack \frac{\sin \left( {\Delta \; \omega \; {T/2}} \right)}{\Delta \; \omega \; {T/2}} \right\rbrack}}}} & (27) \\ {\mspace{20mu} {\rho_{3} = {\frac{V_{1}^{{- \alpha}\; L}}{4}{{{\cos \left( {\theta_{1} - \theta_{2} - {\beta \; L} - {\Delta \; \omega \; {T/2}}} \right)}\left\lbrack \frac{\sin \left( {\Delta \; \omega \; {T/2}} \right)}{\Delta \; \omega \; {T/2}} \right\rbrack}.}}}} & (28) \end{matrix}$

The ratio

$\frac{\sin \left( {\Delta \; \omega \; {T/2}} \right)}{\Delta \; \omega \; {T/2}}$

remains close to unity for small values of ΔωT. Since the two oscillators 612 are very close in frequency, ΔωT<<1 can be achieved.

σ₃ is similarly integrated:

$\begin{matrix} {\sigma_{3} = {{\frac{1}{T}{\int_{- T}^{0}{{t}\left\{ {\frac{1}{4}V_{1}^{{- \alpha}\; L}{\sin \left( {{\Delta \; \omega \; t} + \varphi_{1}} \right)}} \right\}}}} = {\frac{V_{1}^{{- \alpha}\; L}}{4\Delta \; \omega \; T}{\int_{\varphi_{1} - {\Delta \; \omega \; T}}^{\varphi_{1}}{{u}\left\{ {\sin \; u} \right\}}}}}} & (29) \\ {\sigma_{3} = {{\frac{V_{1}^{{- \alpha}\; L}}{4\Delta \; \omega \; T}\left\{ {{\cos \left( {\varphi_{1} - {\Delta \; \omega \; T}} \right)} - {\cos \; \varphi_{1}}} \right\}} = {\frac{V_{1}^{{- \alpha}\; L}}{4\Delta \; \omega \; T}\left\{ {2{\sin \left( {\varphi_{1} - {\Delta \; \omega \; {T/2}}} \right)}{\sin \left( {\Delta \; \omega \; {T/2}} \right)}} \right\}}}} & (30) \\ {\mspace{20mu} {\sigma_{3} = {\frac{V_{1}^{{- \alpha}\; L}}{4}{{\sin \left( {\varphi_{1} - {\Delta \; \omega \; {T/2}}} \right)}\left\lbrack \frac{\sin \left( {\Delta \; \omega \; {T/2}} \right)}{\Delta \; \omega \; {T/2}} \right\rbrack}}}} & (31) \\ {\mspace{20mu} {\sigma_{3} = {\frac{V_{1}^{{- \alpha}\; L}}{4}{{{\sin \left( {\theta_{1} - \theta_{2} - {\beta \; L} - {\Delta \; \omega \; {T/2}}} \right)}\left\lbrack \frac{\sin \left( {\Delta \; \omega \; {T/2}} \right)}{\Delta \; \omega \; {T/2}} \right\rbrack}.}}}} & (32) \end{matrix}$

For δ₂(t) and ε₂(t), embodiments may let φ₂≡θ₁−θ₂+βL, and set the variable of integration to u=Δωt₂, resulting in:

$\begin{matrix} {\delta_{3} = {{\frac{1}{T}{\int_{0}^{T}\ {{t}\left\{ {\frac{1}{4}V_{2}^{{- \alpha}\; L}{\cos \left( {{\Delta \; \omega \; t} + \varphi_{2}} \right)}} \right\}}}} = {\frac{V_{2}^{{- \alpha}\; L}}{4{\Delta\omega}\; T}{\int_{\varphi_{2}}^{\varphi_{2} + {{\Delta\omega}\; T}}\ {{u}\left\{ {\cos \mspace{11mu} u} \right\}}}}}} & (33) \\ {\mspace{79mu} {\delta_{3} = {\frac{V_{2}^{{- \alpha}\; L}}{4}{{\cos \left( {\varphi_{2} + {{\Delta\omega}\; T\text{/}2}} \right)}\left\lbrack \frac{\sin \left( {{\Delta\omega}\; T\text{/}2} \right)}{{\Delta\omega}\; T\text{/}2} \right\rbrack}}}} & (34) \\ {\mspace{79mu} {\delta_{3} = {\frac{V_{2}^{{- \alpha}\; L}}{4}{{\cos \left( {\theta_{1} - \theta_{2} + {\beta \; L} + {{\Delta\omega}\; T\text{/}2}} \right)}\left\lbrack \frac{\sin \left( {{\Delta\omega}\; T\text{/}2} \right)}{{\Delta\omega}\; T\text{/}2} \right\rbrack}}}} & (35) \\ {ɛ_{3} = {{\frac{1}{T}{\int_{0}^{T}\ {{t}\left\{ {{- \frac{1}{4}}V_{2}^{{- \alpha}\; L}{\sin \left( {{{\Delta\omega}\; t} + \varphi_{2}} \right)}} \right\}}}} = {\frac{V_{2}^{{- \alpha}\; L}}{4{\Delta\omega}\; T}{\int_{\varphi_{2}}^{\varphi_{2} + {{\Delta\omega}\; T}}\ {{u}\left\{ {\sin \mspace{11mu} u} \right\}}}}}} & (36) \\ {\mspace{79mu} {ɛ_{3} = {\frac{V_{2}^{{- \alpha}\; L}}{4}\mspace{11mu} {{\sin \left( {\varphi_{2} + {{\Delta\omega}\; T\text{/}2}} \right)}\left\lbrack \frac{\sin \left( {{\Delta\omega}\; T\text{/}2} \right)}{{\Delta\omega}\; T\text{/}2} \right\rbrack}}}} & (37) \\ {\mspace{79mu} {ɛ_{3} = {\frac{V_{2}^{{- \alpha}\; L}}{4}\mspace{11mu} {{{\sin \left( {\theta_{1} - \theta_{2} + {\beta \; L} + {{\Delta\omega}\; T\text{/}2}} \right)}\left\lbrack \frac{\sin \; \left( {{\Delta\omega}\; T\text{/}2} \right)}{{\Delta\omega}\; T\text{/}2} \right\rbrack}.}}}} & (38) \end{matrix}$

Based on the foregoing, embodiments generate α (i.e., the real part of γ) by combining terms ρ₃ and σ₃.

$\begin{matrix} {{{\rho_{3}^{2} + \sigma_{3}^{2}} = {\frac{1}{16}V_{1}^{2}{^{{- 2}\alpha \; L}\left\lbrack \frac{\sin \left( {{\Delta\omega}\; T\text{/}2} \right)}{{\Delta\omega}\; T\text{/}2} \right\rbrack}^{2}}},{{and}\mspace{14mu} {therefore}},} & (39) \\ {\alpha = {{{- \frac{1}{2L}}\ln \left\{ {16\; \frac{\rho_{3}^{2} + \sigma_{3}^{2}}{V_{1}^{2}}} \right\}} + {\frac{1}{2L}\ln {\left\{ \frac{\sin \left( {{\Delta\omega}\; T\text{/}2} \right)}{{\Delta\omega}\; T\text{/}2} \right\}.}}}} & (40) \end{matrix}$

The logarithm involving ΔωT/2 is very small for reasonable values of ΔωT.

Similarly, embodiments may generate a by combining terms δ₃ and 68 ₃.

$\begin{matrix} {\alpha = {{{- \frac{1}{2L}}\ln \left\{ {16\; \frac{\delta_{3}^{2} + ɛ_{3}^{2}}{V_{2}^{2}}} \right\}} + {\frac{1}{2L}\ln \left\{ \frac{\sin \left( {{\Delta\omega}\; T\text{/}2} \right)}{{\Delta\omega}\; T\text{/}2} \right\}}}} & (41) \end{matrix}$

Both δ₃ and ε₃ include the term θ₁−θ₂+βL+ΔωT/2. Compared to ρ₃ and σ₃, the signs of βL and ΔωT/2 change with respect to the phase difference (θ₁−θ₂). Accordingly, embodiments can eliminate the phase difference by combining expressions for the two directions of signal propagation. To determine the imaginary part β of the propagation constant γ, embodiments form the ratios:

$\begin{matrix} {{\frac{\sigma_{3}}{\rho_{3}} = {{\tan \left( {\varphi_{1} - {{\Delta\omega}\; T\text{/}2}} \right)} = {\tan \left( {\theta_{1} - \theta_{2} - {\beta \; L} - {{\Delta\omega}\; T\text{/}2}} \right)}}},{and}} & (42) \\ {\frac{ɛ_{3}}{\delta_{3}} = {{\tan \left( {\varphi_{2} - {{\Delta\omega}\; T\text{/}2}} \right)} = {{\tan \left( {\theta_{1} - \theta_{2} + {\beta \; L} + {{\Delta\omega}\; T\text{/}2}} \right)}.}}} & (43) \end{matrix}$

From the ratios, embodiments compute:

$\begin{matrix} {{{\theta_{1} - \theta_{2} - {\beta \; L} - {{\Delta\omega}\; T\text{/}2}} = {\tan^{- 1}\left( \frac{\sigma_{3}}{\rho_{3}} \right)}}{and}} & (44) \\ {{\theta_{1} - \theta_{2} + {\beta \; L} + {{\Delta\omega}\; T\text{/}2}} = {{\tan^{- 1}\left( \frac{ɛ_{3}}{\delta_{3}} \right)}.}} & (45) \end{matrix}$

Subtracting the two equations, embodiments compute:

$\begin{matrix} {\beta = {{\frac{1}{2L}\left\{ {{\tan^{- 1}\left( \frac{ɛ_{3}}{\delta_{3}} \right)} - {\tan^{- 1}\left( \frac{\sigma_{3}}{\rho_{3}} \right)}} \right\}} - {\frac{{\Delta\omega}\; T}{2\; L}.}}} & \left( {46A} \right) \end{matrix}$

FIG. 7 shows a flow diagram for a method 700 for determining the propagation constant for WDP 118 in accordance with various embodiments. Though depicted sequentially as a matter of convenience, at least some of the actions shown can be performed in a different order and/or performed in parallel. Additionally, some embodiments may perform only some of the actions shown. The operations of the method 700 can be performed by the drilling system 100. In some embodiments, at least some of the operations of the method 700, as well as other operations described herein, can be performed by a processor executing instructions stored in a computer readable medium.

In the method 700, the drill string 108, comprising WDPs 118, is disposed in the borehole 116. Two or more calibration subs 602 are coupled to the drill string 108. The calibration subs 602 cooperatively characterize the WDPs 118 to determine the propagation constant γ. In some embodiments, the calibration subs 602 are included in the WDP repeater subs 132. Other embodiments position the calibration subs 602 at various locations in the drill string 118. The method 700 is described with reference to an embodiment of the WDP repeater sub 132 that includes the calibration sub 602.

In block 702, two repeater subs 132A and 132B are configured to exchange sinusoidal signal transmissions via the WDP 118. The frequencies and phases of the signals to be exchanged are set. Signal frequency and phase may, for example, be set via command from the surface or preprogrammed into the repeater subs 132. The oscillators 612 of the repeater subs 132, which generate the set frequencies, may not generate precisely the same frequencies.

In block 704, a first of two repeater subs 132A transmits sinusoidal signal to the second of the repeater subs 132B via the WDP 118. The first of the repeater subs 132A may be, for example, uphole from the second repeater sub 132B.

In block 706, the second repeater sub 132B receives the sinusoidal signal transmitted by the first repeater sub 132A and splits the received signal into two identical copies. One of the copies is provided to each of two mixers 604 of the second repeater sub 132B. Each mixer 604 mixes the received sinusoidal signal with one of two sinusoidal signals generated by the oscillator 612 of the second repeater sub 132B. The two sinusoidal signals provided by the oscillator 612 of the second repeater sub 132B (one to each mixer 604) are offset in phase by 90°. The mixers 604 produce output signals in accordance with equations (10) and (13).

In block 708, the signals generated by the mixers 604 are filtered by the low pass filters 606. The low pass filters 606 eliminate or reduce high frequency components of the mixer output signals to produce signal outputs in accordance with equations (11) and (14).

In block 710, the low pass filtered signals are integrated over time. Embodiments may perform the integration before or after the filtered signals are digitized by the ND converters 608 in block 712. Embodiments integrate the filtered signals in accordance with equations (21)-(24), (28), (32), (35), and (38). The second repeater sub 132B may transmit the digitized integrated signal to the first repeater 132A or to a processor 610 disposed at the surface or in the drill string 108.

In block 714, the two repeater subs 132 are reconfigured such that the second repeater sub 132B transmits sinusoidal signal to the first repeater sub 132A via the WDP 118. The frequency and phase of the sinusoidal signal transmitted remains unchanged from the setting applied in block 702.

In block 716, the first repeater sub 132A receives the sinusoidal signal transmitted by the second repeater sub 132B and splits the received signal into two identical copies. One of the copies is provided to each of two mixers 604 of the first repeater sub 132A. Each mixer 604 mixes the received sinusoidal signal with a signal generated by the oscillator 612 of the first repeater sub 132A. The two sinusoidal signals provided by the oscillator 612 of the first repeater sub 132A (one to each mixer 604) are offset in phase by 90°. The mixers 604 produce output signals in accordance with equations (16) and (18).

In block 718, the signals generated by the mixers 604 are filtered by the low pass filters 606 of the first repeater sub 132A. The low pass filters 606 of the first repeater sub 132A eliminate or reduce high frequency components of the mixer output signals to produce signal outputs in accordance with equations (19) and (20).

In block 720, the low pass filtered signals are integrated over time. Embodiments may perform the integration before or after the filtered signals are digitized by the A/D converters 608 of the first repeater sub 132A in block 722. Embodiments integrate the filtered signals in accordance with equations (23), (24), (35), and (38). The first repeater sub 132A may transmit the digitized integrated signal to the second repeater 132B or to a processor 610 disposed at the surface or in the drill string 108.

In block 722, the low pass filtered signals are integrated over time. Embodiments may perform the integration before or after the filtered signals are digitized by the A/D converters 608 in block 722.

In block 724 the processor 610 computes the propagation constant of the WDP 118 based on the information provided by the first and second repeater subs 132. The processor 610 computes the propagation constant in accordance with equations (40), (41), and (46A).

The phase difference between the two oscillators 612 may be determined by adding equations (44) and (45):

$\begin{matrix} {{\theta_{1} - \theta_{2}} = {{\frac{1}{2}{\tan^{- 1}\left( \frac{\sigma_{3}}{\rho_{3}} \right)}} + {\frac{1}{2}{{\tan^{- 1}\left( \frac{ɛ_{3}}{\delta_{3}} \right)}.}}}} & \left( {46B} \right) \end{matrix}$

Once the phase difference between the two oscillators 612 has been determined from equation (46B), the phase difference can be set to 0 degrees by adjusting the phase of one or the other oscillator 612. As is well known, synchronizing the phases of two oscillators can be used to synchronize the frequencies of the two oscillators. Two synchronized oscillators can then be used as clocks for measurements requiring accurate timing. An example of a measurement requiring synchronized oscillators is measuring the arrival times of seismic signals at two physically separated locations.

Returning now to the fault monitor 300, when the fault monitor 300 detects a fault in WDP 118, the nature of the fault (whether it is an open, a short, or some other in-between value) and the location of the fault are unknown. The propagation constant γ(ω), the WDP characteristic impedance z(ω) (from measurements before the fault occurs), and the input impedance Z_(IN)(ω) (from measurements after the fault has occurred) are known. The normalized input impedance ζ(ω)=Z_(IN)(ω)/ Z(ω) is also known. These known quantities are complex numbers, and they are functions of frequency, but the distance L_(f) to the fault is a real number and is not a function of frequency. Additionally, if the fault is either an open or a short, then the reflection coefficient Γ is a real number and it is not a function of frequency.

In general, with a WDP system that uses electrical contacts 204 between the drill pipes 118, the soft faults described earlier are also described by a terminating load Z_(t) which is primarily a real number and which is essentially independent of frequency. If the soft fault is due to a high resistance between two contacts 204, then Z_(t) is real and Z_(t)>|Z|. If the soft fault is a low resistance between a contact 204 and the drill collar, then Z_(t) is real and Z_(t)<|Z|. Hence, even for soft faults, according to equation (2) the reflection coefficient Γ is a real number and it is not a function of frequency.

FIG. 8 shows a flow diagram for a method for determining the location of a fault in wired drill pipe in accordance with principles disclosed herein. Though depicted sequentially as a matter of convenience, at least some of the actions shown can be performed in a different order and/or performed in parallel. Additionally, some embodiments may perform only some of the actions shown. The operations of the method 800 may be performed by the fault monitor 300. At least some of the operations of the method 800 can be performed by the processor 308 executing instructions read from a computer-readable medium (e.g., storage 310).

In block 802, the drill string 108 is disposed in the borehole 116. The drill string 108 includes a downhole communication network comprising WDPs 118 and one or more WDP fault monitors 300. Proper operation of the WDPs 118 is verified, for example, by validation of information packets transferred through the WDP or validation of an expected input impedance.

In block 804, the fault monitor 300 determines a propagation constant for the WDPs 118. Some embodiments of the fault monitor 300 may compute the propagation constant as:

$\begin{matrix} {{\gamma (\omega)} = {{{\alpha (\omega)} + {{j\beta}(\omega)}} = {\frac{1}{D}\ln \left\{ {\left( \frac{Z_{P}}{Z + Z_{P}} \right)\left( \frac{Z_{0}}{{Z_{0}\cos \; {h\left( {\gamma_{0}D} \right)}} + {Z_{2}\sin \; {h\left( {\gamma_{0}D} \right)}}} \right)} \right\}}}} & (47) \end{matrix}$

where:

-   D is the length of a joint of the WDP 118, estimated to be the     length of the cable 202 in the joint of WDP 118; -   Z₀ is the known characteristic impedance of the cable 202; -   γ₀ is the known propagation constant of the cable 202; and -   Z_(P), and S_(S) are impedances of the WDP 118 as shown in schematic     diagram of FIG. 9A. The cable 202 is represented by the transmission     line between points 902 and 904. The pair of contacts 204 is     represented by the circuit elements between points 904 and 906. The     pair of contacts 204 represents the box of one joint of WDP 118 and     pin of another joint of WDP 118. The series inductance L and the     series resistance R form the series impedance Z_(S)=R+jωL, while the     shunt capacitance C and the shunt resistance S form the shunt     impedance Z_(P)=S/(1+jωCS). The values for L, R, C, and S may be     obtained from measurements or may be calculated from the known     geometry of the contacts 204 and insulators 208. Alternatively, the     propagation constant γ(ω)=α(ω)+jβ(ω) may be determined with WDPs 118     in the borehole 116 using equations (40, (41), and (46A). For the     purposes of locating a fault, other methods may be used to estimate     the propagation constant by analytical, numerical, or experimental     methods.

For identical joints of WDP 118, the characteristic impedance z for the string of WDPs 118 may be calculated using the principle of translational symmetry. The input impedance Z at point 902 must be the same as the input impedance to the next joint of WDP 118, also represented by Z at point 906. Equating the impedances at points 902 and 906, and incorporating the cable 202 and circuit elements Z_(S) and Z_(P), produces a set of equations that can be solved to obtain the characteristic impedance Z in terms of known quantities. Once Z has been obtained, the voltages, currents, and power levels, can be calculated as well as the propagation constant. An example of the characteristic impedance for a string of WDPs 118 is shown in FIG. 9B. When Γ=0, the input impedance is equal to the characteristic impedance, Z_(IN)(ω)=Z(ω), indicating that the WDP 118 is operating correctly. When Z_(IN)(ω)≠Z(ω), it indicates that a fault has occurred.

In block 806, while drilling, the impedance measurement system 304 measures the input impedance of the wired drill pipes 118 coupled to the fault monitor 300. The impedance measurement system 304 measures the input impedance of the WDPs 118 for a plurality of angular frequencies ω spanning the bandwidth of the WDPs 118. The measurement may be made at least once when a new joint of WDP 118 is added to the drill string 108. The input impedance may be measured for section of WDPs 118 that is separated by fault monitors 300 (e.g., repeater subs 132 that include a fault monitor 300) so that all sections of WDP 118 are characterized.

In block 808, proper operation of the WDPs 118 is verified. The verification may include validating continued telemetry function (e.g., transmitting an information packet through the WDPs 118 and validating that the packet is received without error), and/or that the measured input impedance is within predetermined limits (e.g., limits based on the resolution or random noise of the WDP telemetry system). If the WDPs 118 are operating properly in block 610, then the impedance measurement is periodically repeated in block 606.

If the WDPs 118 are not operating properly in block 810, then fault distance evaluation logic 316 is applied to compute, as shown in equation (7), and record the normalized input impedance in block 812. The measured impedance values may be stored in the sub (e.g., sub 132, 134) for retrieval when the drill string is extracted from the borehole 116.

In block 814, the fault monitor 300 computes the location of the fault. The fault monitor may apply one or a combination of techniques disclosed herein to compute the distance to the fault, where the distance from the fault monitor 300 to the fault identifies the location of the fault. The location determination may be performed at the surface using impedance measurements stored in the sub (e.g., sub 132, 134), or retrieved from the sub that performed the location determination for WDPs 118 uphole of the sub, where a fault prevented transmission of information from the sub. For a fault located downhole of the fault monitor 300, the fault monitor may transmit impedance measurements, and/or location determinations to the surface. Thus, embodiments may employ fault location determinations from both uphole and downhole of the fault to improve location accuracy.

In block 816, the fault monitor 300 has determined the location of the fault to within a few joints of WDP 118. The drill string 108 is extracted from the borehole 116, and the WDP 118 at the determined fault location is removed from the drill string 116 and replaced.

FIG. 10 shows a flow diagram for a method 800 for determining the distance to a fault in wired drill pipes 118 in accordance with principles disclosed herein. Though depicted sequentially as a matter of convenience, at least some of the actions shown can be performed in a different order and/or performed in parallel. Additionally, some embodiments may perform only some of the actions shown. At least some of the operations of the method 1000 can be performed by the processor 308 executing instructions read from a computer-readable medium (e.g., storage 310). The method 1000 may be applied alone or in combination with other fault distance determination disclosed herein to compute the location of a fault in block 614 of the method 800.

In block 1002, the fault monitor 300 has determined that a fault is present in the wired drill pipes 118. The fault location monitor 300 determines whether the fault is an open circuit or a short circuit by analyzing the imaginary part of the measured input impedance at low frequencies.

FIG. 11 shows a plot of real and imaginary parts of wired drill pipe impedance measurement for wired drill pipes 118 including a short circuit in accordance with principles disclosed herein. More specifically, in FIG. 11, the real ζ′(ω) and imaginary ζ″(ω) parts of measured impedance of WDPs 118 are plotted versus frequency for a short circuit (Z_(t)=0 ohm) located 1000 meters from fault monitor 300. The impedances of FIG. 11 are based on equation (7) and typical values for characteristic impedance, phase velocity, and attenuation for WDP 118. In contrast to the relatively constant values for the real and imaginary parts of Z shown in FIG. 9B, ζ′(ω) and ζ″(ω) vary with frequency, indicating that a fault has occurred. Because ζ″>0 at low frequencies (e.g., <50 KHz) in FIG. 11, the fault of FIG. 11 is indicated to be a short circuit.

FIG. 12 shows a plot of real and imaginary parts of wired drill pipe impedance measurement for wired drill pipes 118 including an open circuit in accordance with principles disclosed herein. More specifically, FIG. 12 is a plot of ζ′(ω) and ζ″(ω) for an open circuit (Z_(t)=1000 ohm) located 1000 meters from fault monitor 300. As seen in FIGS. 11-12, an open circuit and a short circuit have different characteristics versus frequency. In particular, at the lowest frequencies the sign of ζ″ indicates whether the fault is a short ζ″>0, or an open ζ″<0. A short circuit looks like an inductive load jωL at low frequencies, while an open circuit looks like a capacitive load −j/(ωC). Because ζ″<0 a at low frequencies (e.g., <50 KHz) in FIG. 12, the fault of FIG. 12 is indicated to be an open circuit.

In block 1004, the fault monitor 300 computes the distance to the fault based on whether the fault is determined to be a short circuit or an open circuit. Equation (7) is inverted to find the apparent distance, L(ω), to the fault.

$\begin{matrix} {{L(\omega)} = {\frac{1}{2{\gamma (\omega)}}\ln \left\{ {\Gamma \; \frac{{\zeta (\omega)} + 1}{{\zeta (\omega)} - 1}} \right\}}} & (48) \end{matrix}$

The propagation constant γ(ω)=α(ω)+jβ(ω) in equation (48) is assumed to be known from measurements or from modeling. The normalized impedance ζ(ω)=ζ′(ω)+ζ″(ω) is obtained using vector voltmeter 406. When there is no fault, the input impedance is primarily real Z_(IN)(ω)=Z(ω)≈40 ohm, for example. Γ is an unknown quantity on the right hand side of equation (48). For the two limiting cases of a short circuit and an open circuit, Γ=±1. Consequently, when the fault is a short,

$\begin{matrix} {{{L(\omega)} = {\frac{1}{2{\gamma (\omega)}}\ln \left\{ \frac{1 + {\zeta (\omega)}}{1 - {\zeta (\omega)}} \right\}}},{and}} & (49) \end{matrix}$

when the fault is an open,

$\begin{matrix} {{L(\omega)} = {\frac{1}{2{\gamma (\omega)}}\ln {\left\{ \frac{{\zeta (\omega)} + 1}{{\zeta (\omega)} - 1} \right\}.}}} & (50) \end{matrix}$

The explicit frequency dependence is shown as a reminder that the estimated distance to the fault can be a function of frequency when measurement errors are present.

Because ζ″>0 at low frequencies in FIG. 11, the fault of FIG. 11 is indicated to be a short circuit. The fault monitor 300 applies the values for ζ(ω) (impedance) and γ(ω) (propagation constant) in equation (49) to compute the distance to the fault.

Similarly, because ζ″<0 at low frequencies in FIG. 12, the fault of FIG. 12 is indicated to be an open circuit. The fault monitor 300 applies the values for ζ(ω) (impedance) and γ(ω) (propagation constant) in equation (50) to compute the distance to the fault. FIGS. 11 and 12 do not have any noise superimposed on the measurements and thus equations (49) and (50) give accurate results for the distances to the faults.

In block 1006, the fault monitor 300 analyzes the computed distance to the fault over frequency, and determines whether the computed distance to the fault remains relatively constant over the frequency range (e.g., the bandwidth of the WDP 118). The natural logarithm of a complex number is multi-valued and has a branch cut along the negative real-axis in the complex plane. In general, the natural logarithm of a complex number returns an imaginary part modulo 2π: ln(re^(jθ))=ln(r)+j(θ+n2π). Hence, care must be taken to choose the correct complex sheet using formulas (48), (49), or (50). Otherwise, incorrect values for L(ω), as indicated by rapid changes in the estimated length versus frequency, may be obtained. If there are substantial changes in the computed distance, then the fault monitor 300 ensures that a suitable value for n in the imaginary part of the logarithm has been chosen, i.e. ln(re^(jθ))=ln(r)+j(θ+n2π).

If noise is present in the measured impedance Z_(IN)(ω) then averaging can improve the accuracy of the distance to a fault. In block 1008, the fault monitor 300 improves the estimate of the distance to the fault by computing an average

${\langle L\rangle} = {\frac{1}{n}{\sum\limits_{i = 1}^{p}\; {L^{\prime}\left( \omega_{i} \right)}}}$

of the distances computed over a selected frequency range {ω₁, ω₂, ω₃, . . . ω_(p)}. Such averaging may be applied to reduce the effects of random noise on the distance determination. As an example of noisy data, random noise with a standard deviation of 1 ohm is added to the input impedance Z_(IN)(ω). A short circuit is located 100 meters below the fault monitor 300. The input impedance before the fault occurs is approximately 40 ohms. The resulting normalized input impedance ζ(ω) with the short is shown in FIG. 13. Since ζ″>0, one can deduce that the fault is a short circuit and therefore equation (49) applies. FIG. 14 shows the inverted distances to the fault versus frequency using equation (49) applied at each frequency. The estimated distances from individual frequencies are very noisy, but the average value for the distance is very close to 100 meters.

FIG. 15 shows a flow diagram for an alternative method 1500 for determining the distance to a fault in wired drill pipes 118 in accordance with principles disclosed herein. Though depicted sequentially as a matter of convenience, at least some of the actions shown can be performed in a different order and/or performed in parallel. Additionally, some embodiments may perform only some of the actions shown. At least some of the operations of the method 1500 can be performed by the processor 308 executing instructions read from a computer-readable medium (e.g., storage 310). The method 1500 may be applied alone or in combination with other fault distance determination disclosed herein to compute the location of a fault in block 614 of the method 800.

In block 1502, the fault monitor 300 has determined that a fault is present in the wired drill pipes 118. The fault monitor 300 determines whether the fault is an open circuit or a short circuit by analyzing the imaginary part of the measured input impedance at low frequencies as explained herein with regard to block 1002 of method 1000.

FIG. 16 is an example of the normalized impedance ζ(ω) where a short is located 500 meters from the fault monitor 300. The input impedance before the fault occurs is Z_(IN)(ω)=Z(ω)≈40 ohm. After the short occurs, the normalized input impedance ζ(ω) is plotted versus frequency in FIG. 16. Random noise with a standard deviation of 1 ohm is present in the impedance data. Since ζ″>0 at low frequencies, one can deduce that the fault is a short circuit and therefore equation (49) applies. At higher frequencies, ζ″ has additional zero crossings. The first zero crossing is near frequency f=100 kHz, a second zero crossing is near f=200 kHz, and a third zero crossing is near f=300 kHz. Such zero crossings contain information about the distance to a fault.

In block 1504, the fault monitor 300 identifies one or more zero crossings in the imaginary part, ζ″(ω), of the normalized impedance of the wired dill pipes 118. Some embodiments of the fault monitor 300 may least squares fit a line to ζ″(ω) about the zero crossing to identify the frequency of the zero crossing. An example of a linear fit to data is shown in FIG. 17. It is sufficient to measure the normalized impedance around such zero crossings to estimate the distance to a fault.

The imaginary part crosses zero, ζ″(ω)=0, at the same frequencies regardless of the load impedance Z_(t). To illustrate why this occurs, let Γ=Γ′+jΓ″ and substitute Γ=Γ′+jΓ″ into equation (7) along with γ(ω)=α(ω)+jβ(ω), producing:

$\begin{matrix} {{{\zeta^{\prime}(\omega)} = \frac{{\exp \left( {2\alpha \; L} \right)} - {{\Gamma }^{2}{\exp \left( {{- 2}\alpha \; L} \right)}}}{{\exp \left( {2\alpha \; L} \right)} + {{\Gamma }^{2}{\exp \left( {{- 2}\alpha \; L} \right)}} - {2\Gamma^{\prime}{\cos \left( {2\beta \; L} \right)}}}},{and}} & (51) \\ {{\zeta^{''}(\omega)} = {- {\frac{{2\Gamma^{\prime}{\sin \left( {2\beta \; L} \right)}} - {2\Gamma^{''}{\cos \left( {2\beta \; L} \right)}}}{{\exp \left( {2\alpha \; L} \right)} + {{\Gamma }^{2}{\exp \left( {{- 2}\alpha \; L} \right)}} - {2\Gamma^{\prime}{\cos \left( {2\beta \; L} \right)}}}.}}} & (52) \end{matrix}$

From equation (52), ζ″(ω)=0 occurs when 2Γ′ sin(2βL)−2Γ″ cos(2βL)=0, or when

tan(2βL)=Γ″/Γ′,   (53)

For WDPs 118 that include electrically conductive contacts 204, the reflection coefficient Γ has a very small imaginary part, i.e. |Γ′|>>|Γ″|. Hence, equation (14) can be approximated as

tan(2βL)=0, and   (54)

When β is known, equation (54) can be solved for the distance to the fault L_(f). There are multiple solutions to equation (54) corresponding to zeros on the tangent function. The fault monitor 300 applies the solutions to equation (54), in block 1506, to compute the estimated distance to the fault:

$\begin{matrix} {{{L_{n} = {\frac{n\; \pi}{2\beta_{n}} = {\frac{n\; \pi \; V_{P}}{2\omega_{n}} = \frac{n\; V_{P}}{4f_{n}}}}},{with}}{{n = 1},2,3,\ldots \mspace{11mu},}} & (55) \end{matrix}$

where V_(P)=V_(P)(ω_(n)) is the phase velocity at the frequency of the zero crossing.

The first zero of ζ″(ω) occurs when n=1; the second zero of ζ″(ω) occurs when n=2, and so on, producing the different estimates L_(n). Note that the solutions given by equation (55) do not require knowledge of α or Γ, only knowledge of β.

In FIG. 17, the first zero crossing for ζ″(ω) is plotted with a least squares fit of a straight line. The straight line intercepts ζ″(ω)=0 at frequency f₁=100.45 kHz and therefore L′₁=V_(P)/(4f₁)=(100.45 kHz)/(4·2.0057·10⁸ m/S)=499.6 m.

In block 1508, the fault monitor 300 averages a plurality of values of L_(n) to improve the estimate of the distance to the fault. Using the zero crossings of ζ″(ω) to determine the distance to a fault provides a very robust distance computation with the additional advantage of requiring data at only a few discrete data points at frequencies surrounding the zero crossing. If the fault monitor 300 determines that the fault is so close to the fault monitor 300 that there are no zero crossings in the bandwidth, then the method 800 may be applied to determine distance to the fault.

FIG. 18 shows a flow diagram for another method 1800 for determining the distance to a fault in wired drill pipes 118 in accordance with principles disclosed herein. Though depicted sequentially as a matter of convenience, at least some of the actions shown can be performed in a different order and/or performed in parallel. Additionally, some embodiments may perform only some of the actions shown. At least some of the operations of the method 1800 can be performed by the processor 308 executing instructions read from a computer-readable medium (e.g., storage 310). The method 1800 may be applied alone or in combination with other fault distance determination disclosed herein to compute the location of a fault in block 614 of the method 800.

In block 1802, the fault monitor 300 has determined that a fault is present in the wired drill pipes 118. The fault location monitor 300 determines whether the fault is an open circuit or a short circuit by analyzing the imaginary part of the measured input impedance at low frequencies as explained herein with regard to block 802 of method 800.

In block 1804, the fault monitor 300 least squares fits the measured impedance measured for the WDPs 118 over a wide frequency range to impedance functions denoted by equations (56)-(57) below. Assuming that the reflection coefficient is a real number, i.e. |Γ′|>>|Γ″|, equations (51) and (52) can be approximated as:

$\begin{matrix} {{{\zeta^{\prime}(\omega)} \approx \frac{{\exp \left( {2\alpha \; L} \right)} - {\left( \Gamma^{\prime} \right)^{2}{\exp \left( {{- 2}\alpha \; L} \right)}}}{{\exp \left( {2\alpha \; L} \right)} + {\left( \Gamma^{\prime} \right)^{2}{\exp \left( {{- 2}\alpha \; L} \right)}} - {2\Gamma^{\prime}{\cos \left( {2\beta \; L} \right)}}}},{and}} & (56) \\ {{\zeta^{''}(\omega)} \approx {- {\frac{2\Gamma^{\prime}{\sin \left( {2\beta \; L} \right)}}{{\exp \left( {2\alpha \; L} \right)} + {\left( \Gamma^{\prime} \right)^{2}{\exp \left( {{- 2}\alpha \; L} \right)}} - {2\Gamma^{\prime}{\cos \left( {2\beta \; L} \right)}}}.}}} & (57) \end{matrix}$

Because α tends to be a slowly varying function of frequency, and because Γ′ can be assumed to be a constant, the frequency dependence is primarily in the terms sin(2βL) and cos(2βL) With a knowledge of α(ω) and β(ω), the measured values for ζ′(ω) and ζ″(ω) can be compared to equations (56) and (57) in a least squares minimization routine to solve for the reflection coefficient Γ′ and for the distance to the fault, L.

In block 1806, the fault monitor 300 determines the distance to the fault.

FIG. 19 is an example of the normalized impedance ζ(ω) where the fault is located 2000 meters from the fault monitor 300. The fault is an open with Z_(t)=1000 ohm. Random noise with a standard deviation of 1 volt is present in the data. Because the fault is distant, there are more zero crossings. The normalized impedance data ζ(ω)=ζ′(ω)+jζ″(ω) are fit to equations (56) and (57) for frequencies up to 100 kHz in FIG. 20. The first zero crossing of the fit to equation (57) for ζ″(ω) occurs at f₁=24.94 kHz. Applying equation (55), L₁=(24.94 kHz)/(4·1.98·10⁸ m/S)=1985 m. The fault is located within 15 meters.

These methods can also be applied to cases where a high series resistance occurs rather than an open circuit, as might occur when there is poor electrical contact between contacts 204. These methods can also be applied when there is a leakage resistance between the contacts 204 and the drill pipe, rather than a short circuit, as might occur if the insulators 208 have failed. In these cases, the magnitude of the reflection coefficient |Γ′| will be less than 1, but these methods are easily extended to cover these cases.

The above discussion is meant to be illustrative of principles and various exemplary embodiments of the present invention. Numerous variations and modifications will become apparent to those skilled in the art once the above disclosure is fully appreciated. For example, while certain embodiments have been described with reference to determining oscillator phase offset, determining a propagation constant, and locating a fault in wired drill pipes, those skilled in the art will understand that embodiments are applicable to locating faults, and/or determining a propagation constant, and/or determining oscillator phase offset in various communication systems that employ suitable communication media. 

What is claimed is:
 1. A method for locating a fault in wired drill pipe, comprising: disposing a drill string comprising a plurality of wired drill pipes in a borehole; measuring the input impedance of the wire drill pipes while drilling: determining a propagation constant for the wire drill pipes; determining, based on the input impedance, whether a fault in the wired drill pipe is an open circuit or a short circuit.
 2. The method of claim 1, wherein the analyzing comprising analyzing an imaginary portion of the input impedance and determining, based on the imaginary part of the input impedance, whether the fault is an open circuit or a short circuit.
 3. The method of claim 2, wherein the analyzing further comprises: determining that the fault is an open circuit based on the imaginary part being negative below a predetermined frequency; and determining that the fault is a short circuit based on the imaginary part being positive below the predetermined frequency.
 4. The method of claim 1, further comprising determining a distance from a point of input impedance measurement to the fault as a function of the propagation constant, the measured input impedance, and whether the fault is an open circuit or a short circuit.
 5. The method of claim 4, wherein determining the distance further comprising averaging a plurality of distances to the fault determined for a number of different frequencies.
 6. The method of claim 1, further comprising: identifying a zero crossing in an imaginary portion of the input impedance; and determining a distance from a point of impedance measurement to the fault as a function of a frequency at the zero crossing and phase velocity at the frequency.
 7. The method of claim 6, wherein the identifying comprises identifying a plurality of zero crossings in the imaginary portion of the input impedance; determining the distance comprises determining a different distance for each of the zero crossings and averaging the different distances.
 8. The method of claim 1, further comprising: least squares fitting the measured input impedance to a function for a real portion of the input impedance and a function for an imaginary portion of the input impedance; and determining a distance from a point of impedance measurement to the fault based on the fitting.
 9. The method of claim 1, further comprising determining a location of the fault based on a first input impedance measured from downhole of the fault in the wired drill pipe.
 10. The method of claim 9, further comprising determining a location of the fault based on a second input impedance measured from uphole of the fault in the wired drill pipe.
 11. Apparatus for drilling a borehole in formations, comprising: a drill string comprising a plurality of wired drill pipes; and a wired drill pipe fault monitor coupled to the wired drill pipes, the fault monitor comprising: an impedance measuring system configured to measure an input impedance of the wired drill pipes while drilling the borehole; and a fault locator configured to: determine a propagation constant for the wired drill pipes; and determine, as a function of the input impedance and the propagation constant, a location of a fault in the wired drill pipes.
 12. The apparatus of claim 11, wherein the fault locator is configured to determine, based on the input impedance whether the fault in the wired drill pipes is an open circuit or a short circuit,
 13. The apparatus of claim 12, wherein the fault locator is further configured to determine the location of the fault in the wired drill pipes based on whether the fault is an open or a short.
 14. The apparatus of claim 12, wherein the fault locator is configured to analyze an imaginary part of the input impedance and determine, based on the imaginary portion of the input impedance whether the fault is an open circuit or a short circuit.
 15. The apparatus of claim 14, wherein the fault locator is configured to: determine that the fault is an open circuit based on the imaginary part being negative at predetermined low frequencies; and determine that the fault is a short circuit based on the imaginary part being positive at predetermined low frequencies.
 16. The apparatus of claim 11, wherein the fault locator is configured to determine a plurality of distances from the impedance measuring system to the fault, each distance corresponding to a different frequency; and average the plurality of distances to obtain a final distance to the fault.
 17. The apparatus of claim 11, wherein the fault locator is configured to: identify a zero crossing in an imaginary portion of the input impedance; and determine a distance from the impedance measuring system to the fault as a function of a frequency at the zero crossing and phase velocity at the frequency.
 18. The apparatus of claim 17, wherein the fault locator is configured to: identify a plurality of zero crossings in the imaginary part of the input impedance; determine a different distance to the fault for each of the zero crossings; and average the different distances to obtain a final distance to the fault.
 19. The apparatus of claim 11, wherein the fault locator is configured to: least squares fit the measured input impedance to a function for a real portion of the input impedance and a function for an imaginary portion of the input impedance; and determine a distance from the impedance measuring system to the fault based on the fit functions.
 20. The apparatus of claim 11, further comprising: a plurality of subs, each sub separated from the next by a plurality of wired drill pipes; wherein each sub comprises an instance of the wired drill pipe fault monitor; and each sub is configured to: measure at least one of an uphole input impedance of wired drill pipes uphole of the repeater and a downhole input impedance of wired drill pipes downhole of the repeater; wherein a computer disposed at the surface is configured to determine the location of the fault in the wired drill pipes between two subs based on the downhole input impedance.
 21. The apparatus of claim 20, wherein the computer is configured to determine the location of the fault in the wired drill pipes between two subs based on both the uphole input impedance and the downhole input impedance.
 22. A fault location system, comprising: a plurality of conductively coupled media sections, each media section comprising: a length of conductive media; and conductive couplers communicatively connected to opposing ends of the conductive media; impedance measurement electronics configured to measure an input impedance of the media sections; and a fault locator configured to: determine a propagation constant for the media sections; analyze the input impedance and determine, as a function of the input impedance and the propagation constant, a location of a fault in the media sections.
 23. The fault location system of claim 22, wherein the media sections are joints of wired drill pipe.
 24. The fault location system of claim 22, wherein the fault locator is configured to: determine, based on the input impedance whether the fault is an open circuit or a short circuit; and determine the location of the fault based on whether the fault is an open or a short.
 25. The fault location system of claim 24, wherein the fault locator is configured to: determine that the fault is an open circuit based on an imaginary part of the input impedance being negative at predetermined low frequencies; and determine that the fault is a short circuit based on the imaginary part being positive at predetermined low frequencies.
 26. The fault location system of claim 22, wherein the fault locator is configured to determine a plurality of distances from the impedance measurement electronics to the fault, each distance corresponding to a different frequency; and average the plurality of distances to obtain a final distance to the fault.
 27. The fault location system of claim 22, wherein the fault locator is configured to: identify a zero crossing in an imaginary portion of the input impedance; and determine a distance from the impedance measurement electronics to the fault as a function of a frequency at the zero crossing and phase velocity at the frequency.
 28. The fault location system of claim 27, wherein the fault locator is configured to: identify a plurality of zero crossings in the imaginary part of the input impedance; determine a different distance to the fault for each of the zero crossings; and average the different distances to obtain a final distance to the fault.
 29. The fault location system of claim 22, wherein the fault locator is configured to: least squares fit the measured input impedance to a function for a real portion of the input impedance and a function for an imaginary portion of the input impedance; and determine a distance from the impedance measurement electronics to the fault based on the fit functions.
 30. The fault location system of claim 22, wherein the fault locator is configured to determine the location of fault based on impedance measurements taken from two sides of the fault.
 31. A channel characterization system, comprising: a first calibration unit, and a second calibration unit coupled to the first calibration unit via a conductive medium; and a processor coupled to the first calibration unit and the second calibration unit; wherein the first and second calibration units are configured to: exchange characterization signals via the conductive medium; measure amplitude and phase of the characterization signal received via the conductive medium from the other calibration unit; and provide the amplitude and phase measurements to the processor; and wherein the processor is configured to determine a propagation constant of the conductive medium based on the measurements.
 32. The system of claim 31, wherein each calibration unit comprises an oscillator configured to: generate a characterization signal having a predetermined frequency for transmission via the conductive medium based on the calibration unit being set to transmit the characterization signal; and based on the calibration unit being set to receive the characterization signal via the conductive medium, to: generate a first comparison signal at the predetermined frequency; and generate a second comparison signal at the predetermined frequency having quadrature phase offset from the first comparison signal.
 33. The system of claim 31, wherein each calibration unit comprises: a first mixer configured to mix the characterization signal received via the conductive medium with a sine signal generated by an oscillator of the calibration unit; and a second mixer configured to mix the characterization signal received via the conductive medium with a cosine signal generated by the oscillator of the calibration unit.
 35. The system of claim 33, further comprising an integrator configured to: integrate output of the first mixer over time; and integrate output of the second mixer over time.
 34. The system of claim 31, wherein each calibration unit comprises: a low pass filter configured to: block a sum of the frequency of the characterization signal received via the conductive medium and a signal generated by an oscillator of the calibration unit; and pass a difference of the frequency of the characterization signal received via the conductive medium and the signal generated by the oscillator of the calibration unit
 35. The system of claim 31, wherein the processor is configured to: determine an imaginary portion of the propagation constant based on the characterization signal transmitted from the first calibration unit to the second calibration unit; and determine a real portion of the propagation constant based on the characterization signal transmitted from the first calibration unit to the second calibration unit and the characterization signal transmitted from the second calibration unit to the first calibration unit.
 36. The system of claim 35, wherein the processor is configured to determine the imaginary portion as a logarithm of a sum of a time integrated sum of the received characterization signal and a locally generated sine signal and a time integrated sum of the received characterization signal and a locally generated cosine signal.
 37. The system of claim 35, wherein the processor is configured to determine the real portion as a difference of an inverse tangent of values derived from the characterization signal transmitted from the first calibration unit to the second calibration unit and an inverse tangent of values derived from the characterization signal transmitted from the second calibration unit to the first calibration unit.
 38. The system of claim 31, wherein conductive medium is wired drill pipe.
 39. A method for characterizing a communication channel, comprising: splitting, by a first calibration unit, a calibration signal transmitted by a second calibration unit via a conductive medium connecting the first and second calibration units into a first two signals; mixing, by the first calibration unit, a first of the first two signals with a first oscillator signal generated by the first calibration unit to produce a first mixed signal; and mixing, by the first calibration unit, a second of the first two signals with a second oscillator signal generated by the first calibration unit to produce a second mixed signal; wherein the first and second oscillator signals generated by the first calibration unit have a same frequency and quadrature phase offset; filtering a sum of the first of the first two signals and the first oscillator signal generated by the first calibration unit from the first mixed signal to produce a first filtered signal; filtering a sum of the second of the first two signals and the second oscillator signal generated by the first calibration unit from the second mixed signal to produce a second filtered signal; integrating the first filtered signal over time to generate a first integrated signal; integrating the second filtered signal over time to generate a second integrated signal; and computing a propagation constant for the conductive medium based on the first and second integrated signals.
 40. The method of claim 39, wherein computing the propagation constant comprises computing an imaginary portion of the propagation constant based on the first and second integrated signals.
 41. The method of claim 39, further comprising: splitting, by the second calibration unit, a calibration signal transmitted by the first calibration unit via the conductive medium into a second two signals; mixing, by the second calibration unit, a first of the second two signals with a first oscillator signal generated by the second calibration unit to produce a third mixed signal; and mixing, by the second calibration unit, a second of the second two signals with a second oscillator signal generated by the second calibration unit to produce a fourth mixed signal; wherein the first and second oscillator signals generated by the second calibration unit have a same frequency and quadrature phase offset; filtering a sum of the first of the second two signals and the first oscillator signal generated by the second calibration unit from the third mixed signal to produce a third filtered signal; filtering a sum of the second of the second two signals and the second oscillator signal generated by the second calibration unit from the fourth mixed signal to produce a fourth filtered signal; integrating the third filtered signal over time to generate a third integrated signal; integrating the fourth filtered signal over time to generate a fourth integrated signal; and computing the propagation constant for the conductive medium based on the third and fourth integrated signals.
 42. The method of claim 41, wherein computing the propagation constant comprises computing an imaginary portion of the propagation constant based on the third and fourth integrated signals.
 43. The system of claim 42, wherein computing the imaginary portion of the propagation constant comprises computing a logarithm of a sum of the third and fourth integrated signals squared.
 44. The method of claim 41, wherein computing the propagation constant comprises computing a real portion of the propagation constant based on the first, second, third and fourth integrated signals.
 45. The method of claim 44, wherein computing the real portion of the propagation constant comprises computing a difference of: an inverse tangent of a ratio of the third and fourth integrated signals, and an inverse tangent of a ratio of the first and second integrated signals.
 46. A phase calibration system, comprising: a first unit and a second unit configured to communicate via a communication medium that can propagate sinusoidal waves and data; and a processor coupled to the first unit and the second unit; wherein each of the first unit and the second unit comprises an oscillator, and is configured to: exchange sinusoidal signals via the communication medium; measure phase of the sinusoidal signals received via the communication medium from the other of the first unit and the second unit; and provide phase measurements to the processor; wherein the processor is configured to determine the phase difference between the two calibration units. 